• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Complex Tetration, to base exp(1/e) Ember Edison Junior Fellow Posts: 16 Threads: 4 Joined: May 2019 05/05/2019, 11:38 PM Hi, I was reading the article[1] and i can't reproduce it in mathematica. I need some help, and very much need some code. Edison [1]https://arxiv.org/abs/1105.4735 sheldonison Long Time Fellow Posts: 614 Threads: 22 Joined: Oct 2008 05/07/2019, 04:17 PM (This post was last modified: 05/07/2019, 04:46 PM by sheldonison.) (05/05/2019, 11:38 PM)Ember Edison Wrote: Hi, I was reading the article[1] and i can't reproduce it in mathematica. I need some help, and very much need some code. Edison [1]https://arxiv.org/abs/1105.4735 Equation 18 is the key, which is the asymptotic ecalle formal power series Abel function for iterating $z\mapsto\exp(z)-1$ which is congruent to iterating $\eta=\exp(1/e);\;\;\;y\mapsto\eta^y;\;\;\;z=\frac{y}{e}-1;$ The asymptotic series for the Abel equation for iterating z is given by equation 18.  I have used this equation to also get the value of Tetration or superfunction for base $\eta=\exp(1/e)$, by using a good initial estimate, and then Newton's method.  If you use pari-gp or are interested in downloading pari-gp, I can post the pari-gp code here, including the logic to generate the formal asymptotic series equation for equation 18, and the logic to generate the two superfunctions and their inverses.   $\alpha(z)=-\frac{2}{z}+\frac{1}{3}\log(\pm z)-\frac{1}{36}z+\frac{1}{540}z^2+\frac{1}{7776}z^3-\frac{71}{435456}z^4+...$ If the the asymptotic series is properly truncated then the Abel function approximation can be superbly accurate.   $\alpha(z)\approx\alpha(\exp(z)-1)+1$ To get arbitrarily accurate results, we iterate $z\mapsto\exp(z)-1$ enough times or for the repellilng flower, we can iterate $z\mapsto\log(z+1)$ enough times so that z is small and the asymptotic series works well.  - Sheldon Ember Edison Junior Fellow Posts: 16 Threads: 4 Joined: May 2019 05/08/2019, 12:25 PM (05/07/2019, 04:17 PM)sheldonison Wrote: (05/05/2019, 11:38 PM)Ember Edison Wrote: Hi, I was reading the article[1] and i can't reproduce it in mathematica. I need some help, and very much need some code. Edison [1]https://arxiv.org/abs/1105.4735 Equation 18 is the key, which is the asymptotic ecalle formal power series Abel function for iterating $z\mapsto\exp(z)-1$ which is congruent to iterating $\eta=\exp(1/e);\;\;\;y\mapsto\eta^y;\;\;\;z=\frac{y}{e}-1;$ The asymptotic series for the Abel equation for iterating z is given by equation 18.  I have used this equation to also get the value of Tetration or superfunction for base $\eta=\exp(1/e)$, by using a good initial estimate, and then Newton's method.  If you use pari-gp or are interested in downloading pari-gp, I can post the pari-gp code here, including the logic to generate the formal asymptotic series equation for equation 18, and the logic to generate the two superfunctions and their inverses.   $\alpha(z)=-\frac{2}{z}+\frac{1}{3}\log(\pm z)-\frac{1}{36}z+\frac{1}{540}z^2+\frac{1}{7776}z^3-\frac{71}{435456}z^4+...$ If the the asymptotic series is properly truncated then the Abel function approximation can be superbly accurate.   $\alpha(z)\approx\alpha(\exp(z)-1)+1$ To get arbitrarily accurate results, we iterate $z\mapsto\exp(z)-1$ enough times or for the repellilng flower, we can iterate $z\mapsto\log(z+1)$ enough times so that z is small and the asymptotic series works well. Yes, I need it! I think just has something wrong when i am definiting function. Source code will be helpful. sheldonison Long Time Fellow Posts: 614 Threads: 22 Joined: Oct 2008 05/08/2019, 04:50 PM (This post was last modified: 05/08/2019, 05:38 PM by sheldonison.) (05/05/2019, 11:38 PM)Ember Edison Wrote: Yes, I need it! I think just has something wrong when i am definiting function. Source code will be helpful.[attachment=1343] Code:\r baseeta.gp initeta();         /* initeta initializes kecalle series; 25terms */ slog1=slogeta(1);  /* renormalize so slog(1)=0; slog1=3.029297214418036; */ z=slogeta(2.5)     /* 21.038456088895745460253062718325504556;    */ ploth(t=-1.5,25,sexpeta(t));  /* plot of sexpeta; sexpeta(0)=1    */ z2=invcheta(100)   /*  0.79336896191958487417879655443666434028   */ z1=invcheta(4);    /* -4.5049005907984782975089673142337641018    */ ploth(t=z1,z2,cheta(t));  /* plot of upper superfucntion of eta   */ z=slogeta(I)       /* -1.217279555798763 + 0.5193692007946583*I   */ z=invcheta(I)      /*  1.808671078843811 + 1.565868985090261*I    */ z=cheta(1+I)       /* -6.501975132474055 + 4.920389603877520*I    */   baseeta.gp (Size: 6.4 KB / Downloads: 19) - Sheldon Ember Edison Junior Fellow Posts: 16 Threads: 4 Joined: May 2019 05/08/2019, 06:20 PM (05/08/2019, 04:50 PM)sheldonison Wrote: (05/05/2019, 11:38 PM)Ember Edison Wrote: Yes, I need it! I think just has something wrong when i am definiting function. Source code will be helpful. Code:\r baseeta.gp initeta();         /* initeta initializes kecalle series; 25terms */ slog1=slogeta(1);  /* renormalize so slog(1)=0; slog1=3.029297214418036; */ z=slogeta(2.5)     /* 21.038456088895745460253062718325504556;    */ ploth(t=-1.5,25,sexpeta(t));  /* plot of sexpeta; sexpeta(0)=1    */ z2=invcheta(100)   /*  0.79336896191958487417879655443666434028   */ z1=invcheta(4);    /* -4.5049005907984782975089673142337641018    */ ploth(t=z1,z2,cheta(t));  /* plot of upper superfucntion of eta   */ z=slogeta(I)       /* -1.217279555798763 + 0.5193692007946583*I   */ z=invcheta(I)      /*  1.808671078843811 + 1.565868985090261*I    */ z=cheta(1+I)       /* -6.501975132474055 + 4.920389603877520*I    */ Thank you! I am reading. « Next Oldest | Next Newest »

 Possibly Related Threads... Thread Author Replies Views Last Post An explicit series for the tetration of a complex height Vladimir Reshetnikov 13 9,593 01/14/2017, 09:09 PM Last Post: Vladimir Reshetnikov Is bounded tetration is analytic in the base argument? JmsNxn 0 1,053 01/02/2017, 06:38 AM Last Post: JmsNxn tetration base sqrt(e) tommy1729 2 2,880 02/14/2015, 12:36 AM Last Post: tommy1729 Explicit formula for the tetration to base $$e^{1/e}$$? mike3 1 2,519 02/13/2015, 02:26 PM Last Post: Gottfried tetration base > exp(2/5) tommy1729 2 2,660 02/11/2015, 12:29 AM Last Post: tommy1729 Negative, Fractional, and Complex Hyperoperations KingDevyn 2 5,478 05/30/2014, 08:19 AM Last Post: MphLee regular tetration base sqrt(2) : an interesting(?) constant 2.76432104 Gottfried 7 7,883 06/25/2013, 01:37 PM Last Post: sheldonison tetration base conversion, and sexp/slog limit equations sheldonison 44 48,340 02/27/2013, 07:05 PM Last Post: sheldonison new results from complex dynamics bo198214 25 30,002 02/25/2013, 01:42 AM Last Post: sheldonison Complex Tetration Balarka Sen 16 19,402 02/21/2013, 02:23 PM Last Post: sheldonison

Users browsing this thread: 1 Guest(s)